SpiceDouble dvsep_c (ConstSpiceDouble s1, ConstSpiceDouble s2 )
Calculate the time derivative of the separation angle between
two input states, S1 and S2.
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
s1 I State vector of the first body
s2 I State vector of the second body
s1 the state vector of the first target body as seen from
s2 the state vector of the second target body as seen from
An implicit assumption exists that both states lie in the same
refrence frame with the same observer for the same epoch. If this
is not the case, the numerical result has no meaning.
The function returns the double precision value of the time derivative
of the angular separation between S1 and S2.
1) The error SPICE(NUMERICOVERFLOW) signals if the inputs S1, S2
define states with an angular separation rate ~ DPMAX().
2) If called in RETURN mode, the return has value 0.
3) Linear dependent position components of S1 and S1 constitutes
a non-error exception. The function returns 0 for this case.
In this discussion, the notation
< V1, V2 >
indicates the dot product of vectors V1 and V2. The notation
V1 x V2
indicates the cross product of vectors V1 and V2.
To start out, note that we need consider only unit vectors,
since the angular separation of any two non-zero vectors
equals the angular separation of the corresponding unit vectors.
Call these vectors U1 and U2; let their velocities be V1 and V2.
For unit vectors having angular separation
|| U1 x U1 || = ||U1|| * ||U2|| * sin(THETA) (1)
|| U1 x U2 || = sin(THETA) (2)
and the identity
| < U1, U2 > | = || U1 || * || U2 || * cos(THETA) (3)
| < U1, U2 > | = cos(THETA) (4)
Since THETA is an angular separation, THETA is in the range
0 : Pi
Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0,
we have for any value of THETA other than 0 or Pi
cos(THETA) = s * ( 1 - sin (THETA) ) (5)
< U1, U2 > = s * ( 1 - sin (THETA) ) (6)
At this point, for any value of THETA other than 0 or Pi,
we can differentiate both sides with respect to time (T)
< U1, V2 > + < V1, U2 > = s * (1/2)(1 - sin (THETA))
* (-2) sin(THETA)*cos(THETA)
* d(THETA)/dT (7a)
Using equation (5), and noting that s = 1/s, we can cancel
the cosine terms on the right hand side
< U1, V2 > + < V1, U2 > = (1/2)(cos(THETA))
* (-2) sin(THETA)*cos(THETA)
* d(THETA)/dT (7b)
With (7b) reducing to
< U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT (8)
Using equation (2) and switching sides, we obtain
|| U1 x U2 || * d(THETA)/dT = - < U1, V2 > - < V1, U2 > (9)
or, provided U1 and U2 are linearly independent,
d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2|| (10)
Note for times when U1 and U2 have angular separation 0 or Pi
radians, the derivative of angular separation with respect to
time doesn't exist. (Consider the graph of angular separation
with respect to time; typically the graph is roughly v-shaped at
the singular points.)
E.D. Wright (JPL)
-CSPICE Version 1.0.0, 09-MAR-2009 (EDW) (NJB)
time derivative of angular separation